Law Sine And Cosine

Using the Law of Sines and Law of Cosines on Real Problems

I keep running into the same issues people have when they first try to apply the Law Sine And Cosine outside of textbook examples. You set up a triangle problem, you pick a formula, you get an answer, and then you realize the problem wasn't what you thought it was. Let me walk through how these laws actually work in practice, the cases where they break down, and the one edge case that cost me a lot of time on a project back in 2019. The Law of Sines relates each side of a triangle to the sine of its opposite angle. It looks like this: a/sin(A) = b/sin(B) = c/sin(C). The Law of Cosines is the more general version. It looks like c² = a² + b² - 2ab·cos(C). The Law of Sines only works cleanly when you have either an angle and its opposite side (the AAS or ASA case) or two sides and a non-included angle (the SSA case, which is where things get messy). The Law of Cosines works when you have three sides (SSS) or two sides with the included angle (SAS). I used to teach these formulas in isolation, which is a mistake. The real skill is knowing which law to reach for and, more importantly, when to stop and check your geometry before plugging numbers in. Here is how I approach a problem now.

Step-by-Step: Picking the Right Law

When you are given three sides of a triangle and asked to find an angle, use the Law of Cosines first. It directly gives you the cosine of the angle you want. Rearrange it to cos(C) = (a² + b² - c²)/(2ab). Plug in your values. Take the inverse cosine. Done. This usually takes about thirty seconds on paper if you are careful with arithmetic. When you are given two sides and the included angle, use the Law of Cosines to find the third side. Then use the Law of Sines to find one of the remaining angles. I recommend finding the smaller angle first using the Law of Sines, because the smaller angle is less likely to introduce ambiguity. Then use the fact that angles sum to 180 degrees for the last angle. The whole process takes about two minutes. When you are given two angles and any side, use the Law of Sines. Find the third angle first by subtracting the known angles from 180. Then set up a proportion: a/sin(A) = b/sin(B). Cross-multiply and solve. This is the simplest case and should take under a minute.

When you are given two sides and a non-included angle, that is the ambiguous case. The Law of Sines will give you a value for sin(B), and since sine is positive in both the first and second quadrants, you get two possible angles. One is acute, one is obtuse. You need to check both. Sometimes both are valid, sometimes only one is, sometimes neither is. This is where most people make errors and where I used to lose points on exams.

The Ambiguous Case in Practice

Let me be blunt about the ambiguous SSA case because nobody explains it clearly. When you use the Law of Sines and get sin(B) = 0.75, your calculator gives you B = 48.6 degrees. But B could also be 131.4 degrees. You have to test both. For each candidate angle, add it to the known angle. If the sum is less than 180, that solution is geometrically possible. If it is greater than 180, that solution is impossible and you discard it. In my experience, about a third of SSA problems have two valid solutions, another third have one, and the rest have none. The none case happens when the given angle is acute, the side opposite that angle is shorter than the other given side, and the calculated sine value for the unknown angle exceeds 1. That means no triangle exists with those measurements. I see this error constantly in homework help threads.

What I Learned the Hard Way

In 2019 I was working on a surveying calculation for a land development project. We had a triangular parcel of land where we knew two sides and a non-included angle. I applied the Law of Sines, found one angle, calculated the rest, and produced boundary measurements. A colleague reviewed the work and pointed out that I had only considered the acute solution and ignored the obtuse possibility. The second configuration produced a triangle with significantly different side lengths and angles. We had to redo the field measurements to determine which configuration was correct. The workaround I use now is simple. Before I finish any SSA calculation, I explicitly write down both possible angles and test each one against the triangle angle sum constraint. If both pass, I acknowledge both solutions. If only one passes, I note why the other failed. If neither passes, I state that no valid triangle exists with the given measurements. This adds about ten seconds to the calculation and has prevented every ambiguous-case error I have made since.

Common Mistakes and How to Avoid Them

The most frequent error is using the Law of Sines when the Law of Cosines would be more appropriate, or vice versa. A quick rule: if you have SAS or SSS, reach for the Law of Cosines. If you have AAS, ASA, or SSA, reach for the Law of Sines. This takes the guesswork out of it. Another common mistake is forgetting to convert your calculator to degree mode. I have seen this cause errors of 30 to 50 percent in final answers. If you are getting values that seem wrong, check your calculator mode before you check your math. A third mistake is assuming the Law of Sines always gives you the correct angle. As I mentioned with the ambiguous case, it does not. The Law of Cosines does not have this problem. When you use cos(C) = (a² + b² - c²)/(2ab), the arccosine function returns a single value between 0 and 180 degrees, which is exactly what you need for a triangle angle. This is why I prefer the Law of Cosines whenever possible, even when the Law of Sines would technically work.

Limitations You Should Know About

Neither law works for right triangles in the same way. For right triangles, basic SOH CAH TOA is faster and less error-prone. The Law of Sines and Law of Cosines will give you the correct answer for a right triangle, but they add unnecessary complexity. Save them for oblique triangles. Both laws assume Euclidean geometry. If you are working on the surface of a sphere, like in navigation or astronomy, you need the spherical versions of these laws. The formulas change significantly, and applying planar versions to spherical problems produces wrong answers that get worse as the triangle gets larger. The Law of Sines becomes numerically unstable when an angle is very close to 0 or 180 degrees. In those cases, the sine values are extremely small, and rounding errors in floating-point arithmetic can produce large relative errors in your final answer. If you are doing this computationally, use the Law of Cosines instead for those edge cases.

A Worked Example

Here is a concrete problem. Triangle ABC has side a = 7, side b = 9, and angle A = 35 degrees. Find all missing parts. First, identify the case. We have two sides and a non-included angle. This is SSA. Use the Law of Sines: sin(B)/b = sin(A)/a. So sin(B) = b·sin(A)/a = 9·sin(35°)/7 = 9·0.5736/7 = 0.7369. B could be 47.5 degrees or 132.5 degrees. Test the first: 35 + 47.5 = 82.5, which is less than 180. Valid. Then C = 180 - 82.5 = 97.5 degrees. Use the Law of Sines again: c = a·sin(C)/sin(A) = 7·sin(97.5°)/sin(35°) = 7·0.9914/0.5736 = 12.09.

Test the second: 35 + 132.5 = 167.5, which is less than 180. Also valid. Then C = 180 - 167.5 = 12.5 degrees. Use the Law of Sines again: c = 7·sin(12.5°)/sin(35°) = 7·0.2164/0.5736 = 2.64. Two valid triangles exist with these measurements. That is the answer. Both are geometrically valid, and both should be reported. If you need a reference sheet or a printable summary of these laws, many educational sites offer free downloadable PDFs. Search for "Law of Sines and Cosines reference sheet" and you will find several. Just verify that the formulas match what I wrote above, because some sources have typos in the Law of Cosines formula, writing it as a² + b² + 2ab·cos(C) instead of minus, which flips the answer completely.